Does anybody know if it's possible for a UK expat working in Qatar to get an Umrah visa that will allow one to drive into Saudi Arabia for Umrah?

Thanks

I think you can get a Omrah visa through a certified Omrah Agent in Qatar, and use your car to travel to the kingdom using the same visa.
Also, check with the KSA embassy in Qatar if it is possible to get the Omrah visa without going through a certified agent.